My Book On Physics

[Æthelwulf]

So, I wasn't sure which subforum would allow me post this, but this is my book on physics I have been writing. I appreciate how difficult physics is when explaining it, thus I also appreciate that my book might require some nip and tucks.

 

I am going to post the second chapter of my book. In each chapter I have a small part dedicated to the mathematics of a certain subject. In this part, the math is concerned with angular momentum, aka quantum spin. I would like some honest criticism on my ability to talk about the subjects. The first and second chapters are really introductions to the nature of the universe and what the universe is made of.

 

So I would love any criticism, thanks!

 

Chapter Two

 

The World of the Infinitesimally Small

 

Atoms, Electrons, Protons and Neutrons... Where does it end?

So now we will take a journey to the world of the quantum. This world is a very small world indeed. In this world, you can fit (in the typical average human) around 7 x 10^27 atoms! An atom, was once believed to be the smallest unit that was physically ''out there.'' Atom's where first proposed by Greek Philosophers who said that the atom would be an indivisible unit.

 

In fact, this is what the word ''atom'' means in Greek. It means ''indivisible.'' It was a Greek by the name of Democritus in approximately 450 BCE who coined the term átomos, which evolved to the modern day root word of atom. The amount of atoms in the average human body is then:

 

7,000,000,000,000,000,000,000,000,000

 

However, what comes next can be quite startling to the non-scientist. As small as this basic unit of matter is, the atom is in fact made of an even smaller collection of particles. These particles are named ''subatomic particles'' for obvious reasons.

 

The most basic kind of atom we know about, is the Hydrogen atom. The Hydrogen atom is the chemical element of Hydrogen. Hydrogen is an odorless, colorless and tasteless substance. Water is made of two hydrogen atoms including one atom of Oxygen.

 

The Hydrogen atom contains two subatomic particles. They are called a proton (which is a positively charged particle) and an electron (which is a negatively charged particle). Together, they make one atomic particle of Hydrogen which will be electrically-neutral because the two charges of the proton and the electron exactly cancel out. Electrons are attracted to the center of the atom under a special force, called the Coulomb Force. The Coulomb Force is an electrostatic force which varies upon the square of the distance. In fact, many laws, such as gravity depend on such an equidistant law where the strength of the force will vary on r^2 from the origin.

 

Hydrogen is in fact the most abundant observable element in the universe as far as we can tell. There are however many more types of elements which have their own atomic structures. And many of these atomic structures will differ that they may contain inside of them. The proton is what is called a 'nucleon' from the root word nucleus, basically meaning that this particle is always found in the center of particles. However, there are some exotic theories suggesting that certain nucleons might be found in a halo-like structure inside the atoms; but this is a little complicated to understand and really not all too revelevant for the discussion here.

 

Particles have what are called half lives and their half life is a measure of how long it will take before they will decay into other particles. However, the proton is an exception it seems. It seems to be infinitely stable! It seems this is the case because protons will not decay into other particles on their own because they are the least energetic particles baryon known. Baryon is simply a name we give to a certain class of particles which is a composite particle consisting of three quarks (we shall come to quarks later).

Another type of nucleon which we have not discussed here and which can be found in the center of many atoms is a neutrons. Neutrons are actually electrically-neutral particles which might have been surmised from it's name.

 

Neutrons actually have a mass which is only a tiny larger than that of a proton. A protons mass is 1,6726 x 10^(-27) kg and the mass of a neutron is 1,6749 x 10^(-27) kg. It differs by quite a little as you can see, using the scientific notation of course. The electron mass is extremely smaller than this. The electrons mass is miniscule in fact 0,00091x10^(-27) kg. In chemistry, it is often taught that because the mass of the electron is so small it is often negligable when calculating the mass of an atom. In fact, you can also speak of the mass of a particle in terms of its MeV (Mega electron Volts). The mass of a neutron is 939.56563 MeV and the mass of a proton is 938.27231 MeV.

 

The number of protons in an atom make the atoms atomic number. It makes up the charge number of the nucleus, which excludes neutrons since neutrons are electrically neutral. The atomic number, most credibly the outermost electrons (in the electron shell, which we will cover soon) is in fact responsible for electro-chemical bonding between other atoms - in short, particles can share electrons and by doing so, they can keep atoms in close range of other atoms and help create a complex structure giving rise to different elements based on their particular bondings.

 

The electron shell is basically the configuration of electrons which can be found in an atom. Because of a very interesting dynamical law of atomic physics, no electron can be found to occupy a single electron energy level with another electron (certain ways to avoid violating this law) is by a very special process called quantum spin, which will be discussed very soon.

 

The closest shell to the nucleus is called the "1 shell" (also called "K shell"), followed by the "2 shell" (or "L shell"), then the "3 shell" (or "M shell") and so on. Each shell can only allow a certain amount of electrons. The first shell can hold up to two electrons, the second shell can hold up to eight electrons and again, so on. Electron shells have quite a lot of detail to them which cannot be covered in it's entirety here, for instance, electrons in the outer shell will have a higher energy and will travel farther from the nucleus than those in inner shells. In fact, before the advent of quantum mechanics (the modern theory) we used to believe that electrons would orbit the nucleus much like planets would orbit round their galactic stars. It turned out that this was not the case and the reason for this will be covered in this book in the chapter, The Quantum Wave Function - but to explain in very simple terms, electron's do not follow defined paths in spacetime. They are actually smeared over all possible paths they might take.

The idea that a particle like an electron could not move around a single path in an atom was in fact a very important factor for quantum field theory. Before quantum field theory, there was a serious problem concerning why electron's simply did not radiate away energy and fall into the nucleus of the atom. Because classical field theory treated an electron with a definate path and position, the electron has a negative charge and should be attracted to the positive charge of the atom which was produced by the proton.

 

Because of this attraction, classical field theory would have believed that the electron would have spiralled into the nuceus causing a devistating effect. The atom would collapse in a tremendous flash of energy!

This is why quantum field theory and it's postulate of the wave function became increasingly popular, along with other reasons. The electron did not in fact have a specific path or a specific position anywhere inside an atom. It did not orbit the nucleus like a planet, but in fact was a product of probability, the measure of the possible states itself of where it might be. This saved the idea that the electron would eventually radiate away energy and fall into the nucleus of atoms; but there was an addition to this postulate as well, known as the Heisenberg Uncertainty Principle.

 

In much the same respect, this principle created by physicist Werner Heisenberg in 1926 stated that no particle (not bound to electrons alone) have definate locations or trajectories simultaneously. In other words, the Uncertainty principle says that either the location or the momentum of a quantum particle can be known as precisely as one wanted, but as one of these quantities is specified more precisely, the value of the other becomes increasingly indeterminate!

 

This was a fundamental property of all matter. It has been tested to a high degree and was part and parcel of the reason why electrons simly could not fall into the nucleus of atomic objects.

Particles like the electron and the proton are in a certain class of family. They are called Fermions. Particles which are called Fermions are in fact spin 1/2 particles. Spin is a very interesting subject which we should cover. However, before I begin to cover some of the mathematics of spin, I first want to give an explanation to what it is, or atleast what we believe it is.

 

The classical electron, is believed to be a sphere with a radius of (e²/Mc²). This is not a measured value. It's a careful analysis of the dimensions of the equation of the radius which says it depends on the electron charge (squared) e² the mass M and the speed of light (squared) c². The quantity in the denominator Mc² actually makes up the rest energy of a particle E_0. The rest energy will be explained in greater detail in the Chapter discussing relativity.

 

The Classical Electron Radius is in fact 1/137 times larger than the Compton Wavelength. The Compton Wavelength is (h/Mc) where h is Plancks Constant and it has a value of 6.62606957(29)×10^(−34) j.s. The Compton Wavelength itself has a value for the electron as 2.4263102175±33×10^(−12) m (the value varies with different particles) and is a measure itself of the wavelength of a particle being equal to a photon (a particle of light energy) whose energy is the same as the rest-mass energy of the particle. Complicated? Yes it can be.

 

Basically, all particles have a wavelength. Photon's can never be at rest (again reasons why will be given in the relativity chapter), but the energy of a photon can be low enough to have it's wavelength match any particle who is at rest. It's often seen in the eye's of many scientists as the ''size'' of a particle. Actually, a more accurate representation of the size of an object would be the Reduced Compton Wavelength (reasons given in the chapter references under [1] ). This is just when you divide the Compton Wavelength by 2π and it gives a smaller representation for the mass of a system.

 

Now, going back to the electron, the electron as a sphere was accepted by most physicists until the age of the revolutionary quantum field theory. A physicist by the name of Wolfgang Pauli actually predicted a very strange property of all subatomic matter. Using careful experiments, he was able to deduct that particles behaved as though they possessed a spin, just like a spinning top or even better to imagine, the spinning surface of a planet like the Earth.

 

Wolfgang didn't actually name it spin however, that was later coined by Ralph Kronig, George Uhlenbeck, and Samuel Goudsmit in 1925 and then a mathematical theory was developed by Pauli in 1927. The classical idea of spin came from Noether's (a mathematician) ''generator of rotations'' where it describes a real physical rotation of quantum objects.

 

This idea soon ran into problems however, because modern quantum field theory did not really believe an electron was classical at all, possessing a radius. In fact, as far as physicists can tell, any attempt at measuring a radius for an electron failed and all experimental data seemed to suggest that the electron was really a pointlike system - that is , a particle which does not have any dimensions which we can obtain from the classical radius. It was just a point and thus some problems began to arise from the mathematical theory.

 

A classical radial system can indeed rotate like a planet. In fact, an electron under this theory could make a 360 degree turn in space and as would be expected would return to it's original orientation. However, for a pointlike system to achieve a rotation and return to it's original orientation it would need to make twice as much as this angle (720 degrees). See, pointlike systems when in accord to rotations don't act like classical radial systems.

 

So it seemed there was a problem. Either particles do not physically rotate yet still possess an angular momentum with a classical radius, or that quantum particles don't physically rotate but still have a quantum spin and intrinsic property no less. It was decided that spin could not be a real physical spin [2]. Instead, the classical electron would fall into the archives of physical curiosity and the idea that spin was some inherent, instrinsic and fundamental property would live on. I believe this might have been a mistake. The fact that we haven't been able to observe the radius as of yet should not indicate an electron has no size. In fact, scientists have just recently attempted to measure the shape of an electron by measuring it's wobble in a magnetic field. An excellent article which has proposed the experiment can be found in the references of this chapter.

 

But, as much to my frustration and to many other's, our theory developed to satisfy the contention that electrons did not have a radius (pointlike systems are tremendously simpler to deal with than three-dimensional systems... this maybe one reason quantum field theory took this course). Nevertheless, an interesting factor worth mentioning is that it changes the mathematics in the transition from classical to modern theory by very little. Spin is so important in physics, that is defines not only the behaviour conducted in experimental physics, but also defines the chemical elements itself. Atoms can have more than one electron - in fact, in a more precise statement, no two electrons in a single atom can have the same four quantum numbers; if n, l, and m_l are the same, m_s must be different. This difference is achieved by having different spin states. Spin states can take either an ''up'' or ''down'' spin. One strange fact of quantum mechanics (which will be covered in the quantum wave function part) is that before any state of either ''up'' or ''down'' has been measured, it can result in a superpositioning of states. This inexorably means that a particle can have both a spin ''up'' and a spin ''down'' simultaneously.

 

A Brief Overview of the Mathematical Application of Spin

 

Angular momentum is a vector quantity, it points in a particular direction. The spin as a vector points in the angle of direction. There are two kinds of generators of rotations we can speak about. There is J which is the total angular momentum which generates complete rotations and then there is L which is the orbital angular momentum which moves the system on circles, so it will move round the origin.

We will introduce now commutation relationships

 

[J_i, J_j] = ∑_k ε_ijk J_k [1]

 

it is also true that

 

[L_i, L_k] = ∑_k ε_ijk L_k [2]

 

The symbol ε_ijk is called the antisymmetric Levi-Cevita Symbol. Canonical commutation relations simply have the identity [X,Y] = XY − YX. To understand a canonical commutation relation, it is the relation between canonical conjugate quantities. One well-known example in physics of two commutation relations in position and momentum, which is expressed here in what is called a Poissen Bracket

 

{x,p} = 1

 

Going back to our relations [1] and [2] we have the following:

 

J² has eigenvalues j(j+1) where j=(0, ½, 1...) for [1]

 

L(L+1)=(0, 1,2...) so only integers allowed for [2]

 

Now we should introduce the fact that

 

S_i ≡ J_i - L_i [3]

 

But what does taking the difference of the total angular momentum and the orbital angular momentum? It just means that S_i will become the generator of rotations for a particle around it's own axis which means we won't be moving the object in this expression.

 

So we can do some really cool stuff with this expression.

 

[s_i, S_j] = [J_i - L_i, J_j - L_j] [4]

 

To express how all the identities here commute with each other can be given as

 

= ∑_k ε_ijk (J_k + L_k - L_k - L_k) [5]

 

The J_i will commute with J_j is what gives us our J_k. L_i with L_j gives L_k. J_i on L_k gives a minus L_k. Another minus L_k appears, but justification would just take more time to explain. The reason why it would be pointless really to explain why is because after all the mathematics, all the L_k signs collapses to a single L_k anyway.

 

Then we simply substitute (J_k - L_k ) found in our equation [5] for S_k from [3] (hopefully noticing that S_i is not expressed with the subscript ''i'' but ''k'' instead to make

 

= i ∑_k ε_ijk S_k [6]

 

Now, there may have been quite a few things in there you have never heard of, like a Cevi-Levita Symbol, Poissen Brackets or Eigenvalues, but I cannot explain everything unfortunately (I will be able to cover Eigenvalues shortly). Hopefully I can give a brief overview of the mathematics and maybe even present it in such a way that it can be easily understood.

 

Now, remember when I said that a man called Pauli created a mathematical theory of spin? Well he did so using three special matrices called Pauli Matrices, which would help explain the spin of any quantum mechanical particle.

 

The Pauli Matrices are what are called by Mathematicians and Physicists as Hermitian Matrices. To physicists, Hermitian Matrices are basically the things which we can observe, which are named Observables. The diagonal components of a matrix are real and to understand the following will require a little knowledge on matrices.

 

As was stated there are three Pauli Matrices given as

 

0 1 = σ(1)

1 0

 

This matrix is Sigma 1.

 

0 -i = σ(2)

-i 0

 

This is sigma 2.

 

1 0 = σ(3)

0 -1

 

And the final matrix as you probably could have surmised is sigma 3, the final Pauli Matrix. Just a quick summery, σ(3) has both an eigenvalue of +1 and -1. An Eigenvalue is attached to the definition of a linear system of equations; vectors are expressed in what is called a Hilbert Space which allows you to compute the configuration of system, usually this is called a configuration of states and physical observables are given by operators. Moreover, the values that the observables take are given by their Eigenvalues. This is purely a postulate of quantum physics.

 

The Pauli matrices are Hermitian matrices as has been noted, but what makes a matrix Hermitian? Well, the matrix requires two things. It needs to be symmetric and real. A symmetric matrix is a square matrix (as most matrices are) that is equal to it's Transpose. If M is some matrix, then if M is symmetric then

 

M^T = M

 

Where for notational purposes, the ''T'' on M indicates it is the Transpose. To make it Hermitian, you need to Transpose the matrix but then you must complex conjugate everything. Doing so, one can state the matrix as

 

M = M†

 

Where the conjugate transpose is given is as M†. The most simplest case of complex conjugates, is given by a complex number z = a + ib where ''a'' and ''b'' are real numbers, the conjugate is z† = a - ib.

Consider the unit vectors n_1 + n_2 + n_3 which incidently equals 1. Then we can have

 

σ∙n

 

which is the ''dot product''. It will allow us to express σ∙n as

 

= (σ∙n)_1 + (σ∙n)_2 + (σ∙n)_3

 

But what does σ∙n physically mean? Well, σ∙n is the basically and very simply, is the component of spin along the direction of n. In fact

 

σ∙n = (σ∙n)_1 + (σ∙n)_2 + (σ∙n)_3

 

Is also Hermitian. A neat way to write this is as

 

n_3 n(-)

n(+) -n_3

 

We obtain n(-) and n(+) from these equations

 

n_1 - in_2 = n(-)

 

and

 

n_1 + in_2 = n(+)

 

There are many things we could have potentially covered concerning this topic but which I have no room to mention. The application of quantum angular momentum is vast with many topics which could be discussed. Hopefully I have given some kind of taster in what quantum mechanics has to say about this area.

 

Later, we will see some fascinating predictions by quantum mechanics and spin, one which is called quantum entanglement. Entanglement is such a deep subject in physics, so we won't discuss it here. I will however briefly mention that it involves a classical proof by John Bell, called Bells Inequality. You can actually talk about Bells Inequality in terms of set theory, which is not an algebra but can form the analog of an algebra.

 

Three circles which are symmetrically touching and perhaps even overlapping and with each compartment specially numbered, you can use the set theory to describe an inequality which reads:

 

N(A and not in B ) + N(B and not C) ≥ N(A and not C)

 

Which has three properties, defined as A, B and C. These can be any properties. N stands for the ''number of entities,'' and then it reads off that ''the number of properties in A and not B plus the number of properties in B and not C is greater or equal to the number of properties in A and not C.''

The fact of mentioning this, is because Spin can be defined two different ways. In Bells original paper, he treated the system with one spin axis. Indeed, a new realm of quantum investigation challenges that proposal using two axes which will define spin 1/2 particles.

 

Violation of Bells Inequality which can be found in quantum mechanics gives an experimental varification of spin 1/2 particles. It may also be indicating that particles are not pointlike at all. A one-dimensional spin state does not have any structure, but a two spin axial particle can with what is called a spin microstructure. The violation is achieved when understanding that the two simultaneous axes form a resonance state which is also known as the spin fringe. [3]

 

The spin microframe is simply related to the ordinary coordinate frame by a simple rotation in space. [4] It is an interesting fact also to mention that it is impossible to distinguish the 1 dimensional spin and the 2 dimensional spin when a magnetic field is present. Particles will spin a particular direction when placed into a magnetic field. [5] I will leave you with one final interesting fact of quantum mechanical spin and the fundamental nature of particles. There has been some composite particles which have no spin at all, called zero-spin particles. However, as interesting fact is that all fundamental quantum particles possess non-zero spins.

 

And it gets even smaller!

 

And particles, as small as we have been able to cover, now get's even smaller. We don't just have protons and neutrons, but we have even smaller particles. A proton and neutron are composite particles, made from what are called quarks. An interesting fact of nature, is that you don't seem to find quarks on their own either. Quarks when created, are always created in like company which give rise to the B-Meson Family which is interesting, as it would make them composite in a sense. [6] The quark comes in six types, called Flavors. Those Flavors are up, down, strange, charm, bottom, and top.

 

The up and down quarks are known to have the smallest masses and the heavier quarks (simply meaning they have a greater mass) tend to be shortlived and will decay into other quark particles.

Quarks are the only particles known which interact with all the forces in nature, those being the Gravitational, Electromagnetic, Strong and Weak forces. The Gravitational force should be recognizable. It is the force which keeps us attracted to the Earth's center. The reason we don't pass right through the earth however, we can thank the electromagnetic force for that. The reason why is because the electrons in your body and the electrons making your chair act like tiny little magnets, give rise to an electrostatic repulsion. Now the Strong and Weak forces are basically nuclear in nature. The weak force is responsible for the decay of a particle into another particle. However the strong force is one of the more interesting forces to consider for this part, since it is mediated by a particle called the Gluon.

 

The Gluon is what is often called ''the quantum glue'' which holds particle's very tight together. Quarks actually exchange this stuff all the time. There are eight types of gluons which are defined by calling these colours. We don't mean that there is actually any colour to these sticky particles, just like we don't infer that quarks have any flavor to them.

 

Gluons could also come in a kind of exotic matter, called Glueballs. These particles are composite particles made from entirely gluon energy. Whilst theoretically they should be allowed to exist in nature, they would be very hard to detect because they will mix easily with ordinary matter states.

 

As far as we know, matter is not made up of any smaller units than these and so that is us; we have taken a journey from the small (atoms) to the fantastically small (quarks) and now we shall venture into the world of Quantum Mechanics, the Bigger Picture.

 

REFERENCES

 

Why Doesn't the Electron Fall Into the Nucleus? Franklin Mason and Robert Richardson, J Chem. Ed. 1983 (40-42)

On the Einstein Podolsky Rosen Paradox by John S. Bell http://www.drchinese.com/David/Bell_Compact.pdf (online paper)

Electrons are spherical: What's round and measures a billionth of a millimetre? http://www.thenakedscientists.com/HTML/content/news-archive/news/2277/

[1] The Compton Wavelength depends on the Planck Constant given as h. However, there is something called a reduced Planck's Constant. The reduced Plancks Constant is ħ and is obtained from dividing Plancks Constant h by 2π (here π is just pi) and it gives a ''smaller'' value of the original Constant.

[2] I've just attempted to explain that whether or not physical particles have a radius, spin might still be an intrinsic effect. We have had to deal with many aspects of quantum theory which seem unusual. The idea that a system might still have a radius but spin remaining as an intrinsic property should not be outside the realm of possibilities. Though this has never been suggested before as far as I am aware. However the idea of completely pointlike systems may still trouble some quiet physicists and as far as I am aware, many non-scientists are indeed troubled by such a thought that something can contain a mass and not a volume. The classical condition of mass is dependant on volume and density.

[3] http://arxiv.org/ftp/arxiv/papers/0707/0707.1763.pdf

[4] This realm of physics using the spin microframe deals with taking the sum of the vectors. In example, you might have

 

μ_z Z + μ_x X = √2 (μn)

 

where n is the unit vector, we can see that you calculate twice the magnitude along a particular direction. We haven't spoke about the magnetic moment and unlikely to in this book.

[5] http://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment

[6] http://press.web.cern.ch/press/PressReleases/Releases2011/PR16.11E.html

[Æthelwulf]

If I had a particular criticism of my own work, I think I should include how the Pauli Matrices where derived... any takes?